Numerical model for light propagation and light intensity

distribution inside coated fused silica capillaries

Tomasz Piasecki ,MirekMacka , BrettPaull , DermotBrabazon

Abstract

Numerical simulations of light propagation through capillaries have been reported to a

limited extent in the literature for uses such as flow-cell design. These have been restricted to

prediction of light path for very specific cases to date. In this paper, a new numerical model

of light propagation through multi-walled cylindrical systems, to represent coated and

uncoated capillaries is presented. This model allows for light ray paths and light intensity

distribution within the capillary to be predicted. Macro-scale (using PMMA and PC

cylinders) and micro-scale (using PTFE coated fused silica capillaries) experiments were

conducted to validate the model's accuracy. These experimental validations have shown

encouragingly good agreement between theoretical predictions and measured results, which

could allow for optimisation of associated regions for monolith synthesis and use in fluidic

chromatography, optical detection systems and flow cells for capillary electrophoresis and

flow injection analysis.

Keywords: Light propagation Light intensity Modelling Capillary Optical detection

1. Introduction

Organic monolithic stationary phases, since their first appearance in 1992, are considered one

of the milestones in chromatography [1,2]. Several methods of monolith polymerisation

initiation have been reported to date including by application of resistance heating, UV-

irradiation, microwaves, electron beam and 7-radiation [3-6]. Photoinitiatedpolymerisation

for monolith formation was successfully demonstrated for the first time in 1997 by Viklund et

al. [3]. From all of these methods, heat and UV-irradiation are the most popular [7]. Better

spatial control of monolith formation is possible with photoinitiatedpolymerisation than other

techniques including thermal polymerisation. Recently it has been demonstrated that Light

Emitting Diodes (LEDs) with wavelengths in the UV and visible region can be used as light

sources for successful photoinitiatedpolymerisation of monoliths [7-9]. LEDs are an

interesting alternative to traditional (incandescent and fluorescent) light sources having low

cost, long life and unique spectral properties. Typical vessels for monoliths are fused silica

capillary based microfluidic chips.

Whenever light is incident on the boundary of two transparent dielectrics, part of it is

reflected and part is transmitted, see Fig. 1.

Angle of incidence is related with angle of transmittance by Snell's Law (Eq. (1)). In the case

of cylindrical symmetry, as in capillaries, it is possible that the exiting ray will not be parallel

with the incident, see Fig. 1.

nisin i= ntsin t

whereni is the refractive index of the incident ray transmission medium, iis the angle

between incident wavevector and the normal to surface, nt is the refractive index of the

material through which the transmitted ray passes and tis the angle between transmitted

wavevector and normal to the surface. Light attenuation inside the absorbing medium is

governed by the Lambert-Beer law (Eq. (2)).

I = Io

Where I is the light intensity at position I, which represents distance of passage through the

medium, Io is the initial light intensity, e is the Euler's number, is the molar absorbance

coefficient and c is the molar concentration.

Numerical modelling of physical processes has become more popular as computing power,

that is readily available, has increased in recent years. Light propagation and related

phenomena are often simulated numerically. Due to the often required problem definition

complexity these simulations typically involve methods of wave optics. Numerical

simulations of light propagation through capillaries have been reported, but as a tool for flow-

cell design and were limited to prediction of light path for these specific cases only [10,11].

This paper presents a developed numerical model for light propagation through multi-layered

systems with cylindrical sym-metry. The model was developed to calculate light propagation

and light intensity distribution within the capillaries. Macro- and micro-scale experiments

were conducted to validate the model's accuracy. Such a model could be used to understand

and optimise monolith synthesis within capillaries. Knowledge of light distribution within

capillaries of various geometries, wall thickness, diameters and with different numbers of

layers could allow for optimisation of associated regions of monolith synthesis. The

presented model can also be used for designing optical detection systems and flow cells for

capillary electrophoresis, liquid chromatography and flow injection analysis.

2. Experimental materials and methods

To evaluate the quality of the numerical predictions two separate sets of experiments were

designed. In the first set, macro-scale experiments with cylinders of PMMA and PC were

illuminated with a 532 nm wavelength green laser diode, see Fig. 2. The polymer tubes made

of polymethylmetacrylate (PMMA) and polycarbonate (PC) were purchased from Access

Plastic company (Dublin, Ireland). The two sets of concentric plastic cylinders were made

with outer and inner diameter as follows: set A: 80 mm x 74 mm PC and 70 mm x 60 mm

PMMAand set B: 80 mm x 70 mm PMMA and 70 mm x 64 mm PC. The refractive indices

used in the developed models for PMMA and PC were 1.495 and 1.592 respectively [12-14].

The macro-scale capillary experimental simulation was mounted on an optical bench. The

green laser of 5 mW with wavelength of 532 nm, which was used as the light source was

mounted on a micro-metric stage pointing parallel to the axis of symmetry (y-axis) as shown

in Fig. 2. The laser was aligned to shine exactly along the longitudal axis of symmetry of the

tube and then moved 40 mm along the x-axis such that the laser line formed a tangent to the

outer cylinder. During the experiment the laser was moved by 1 mm increments toward the

cylinder's centre. At each increment a picture, taken with a Panasonic LUMIX DMC-FZ30

digital camera mounted vertically overhead, was taken to measure the reflection and

transmission angles in each cylinder for comparison with the corresponding numerical model

results. Measurements were taken for points between 15 and 40 mm from the central axis

point. These tests were repeated three times with average reflectance and transmission angles

being recorded. Confidence intervals for these results using a 95% level and t-distribution

were calculated.

In the second set of experiments, micro-scale experimental measurements of the optical light

intensity in a 100 p.m internal diameter capillary were recorded. These measurements were

com-pared with the theoretical values as determined from numerical simulations. The

capillary of 1 cm length was filled with 0.01 M solution of tartrazine and illuminated with

white and 430 nm violet LEDs.

Tartrazine is a yellow food dye with the maximum absorbance for which was measured

around 425 nm, see Fig. 3. Absorbance and emissions spectra were measured using Ocean

Optics MAYAProfibre optic spectrophotometer with SpectraSuiteTM software. The used

Polymicro Technologies transparent PTFE coated fused silica capillaries of 100 jam internal

diameter were purchased from Composite Metal Services Ltd (UK). The LED (type LED430-

06) of 5 mm diameter and 430 nm central wavelength violet was purchased from

RoithnerLasertechnik, GmbH, Vienna, Austria. The white LED of 5 mm diameter was

manufactured by Nichia (type NSPW500GS-K1), was purchased from Dotlight, Germany.

Tartrazine (CAS number 1934-21-0) was purchased from Sigma-Aldrich, Ireland.

The experimental set-up is presented in Fig. 6. Capillary pictures were taken using an

Olympus BH-2 BHSP microscope fitted with a 20 x magnifying lens and an Imagingsource

1.2 megapixel USB camera. Light intensity false-colour maps were generated using Adobe®

Photoshop Extended CS3 and angles were measured in scaled images processed with Image)

1.43u software. Sample statistics were calculated in IBM ii; SPSS ver. 17 statistical software

package.

3. Theory and model set-up

The numerical modelling software was developed to calculate the light ray path through

multi-layered cylinders, and the light intensity distribution map through the cylinder cross-

sectional area. Light propagation within multimode optical fibres occurs by the phenomena of

total internal reflection, where the values of refractive indices and fibre diameter remain

within the limits of geometrical optics. The size of the capillary used in this work was

comparable with the size of multimode optical fibres. It was assumed that capillary body,

coating and bore were perfectly cylindrical and concentric. A second assumption was that the

incident light had the form of parallel rays (spatially collimated), similar to laser beam light.

Only the right half of the capillary cross-section is displayed in the developed model, as the

diameter acts as the modelled axis of symmetry and no light ray could propagate through

from left to right side. In general such occurrence is possible, but only for higher values of

refractive indices approximately twice those of glass and polytetrafluor-oethylene (PTFE),

which were used in this work.

The programmed model calculated the light ray path equa-tions in the Cartesian coordinate

system. Separate linear functions to describe each of the light ray path segments were used

(for example between air/tube, tube/tube or tube/liquid). Each light path segment was

calculated in a separate subroutine calculating the light path in each zone. Incident light was

in the form of rays Parallel to y-axis, see Fig. 4.

The first subroutine calculated the coordinates of the light incident from infinity on the

air/coating boundary. This point of incidence of the ray a1 on outermost circle c1 was

assignedas p1. An extended radius r1 going from(0,0) throughp1was drawn for calculation of

angle of incidence and in turn angle of transmission was calculated from nisin =

ntsin , for the use inputted values of refractive indices ni and nt. A line a2 through p1 was

drawn representing the refracted ray in the coating with as the angle between a2 and r1,

ending the first programme subroutine, see Fig. 4a. The next subroutine began with

calculation of point p2 (where line a2 crossed boundary c2) and the drawing of extended

radius r2 from (0,0) through p2. Angle of incidence θi2 was calculated as the angle bounded by

r2 and a2, see Fig. 4b. Angle of transmittance was calculated from Eq. (1) and the line a3

was drawn where was an angle between r2 and a3, ending the second programme

subroutine, see Fig. 4b. This subroutine was iterated a further three times to calculate light

ray paths segments along lines a4, a5 and a6 after refraction on each encountered boundary.

These next subroutine es of the light path formation are illustrated in Fig. 4c-e and the

complete generated light path is presented in Fig. 4f without reference lines. The entire light

path was represented as a sum of individual rays calculated separately according to the

symbolic algorithm:

where r is the light ray path, k is the step number and n is the number of layers; a, c, p, r,

and are as described earlier.

For light intensity calculations the entire capillary bore (100 m inner diameter, which was

modelled) was divided into a 0.1 m x 0.1 m grid and assigned with initial light intensity

values of zero, see Fig. 5.

This resolution was determined to be adequate giving 500 cells along capillary radius and

1000 cells across capillary diameter. The external capillary surface was illuminated by a set

of parallel light rays spaced 0.1 µm apart. Each individual light ray was propagated through

the entire capillary, giving a single light path for each incident light ray at each x-value.

Modelled light ray incidence direction and x-axis direction are the same as sown in Figs. 4

and 5. Note a new origin (0,0) and y-axis direction are shown in Fig. 5. For light intensity

modelling it was assumed that there was no light absorption or attenuation in the capillary

coating and capillary wall. Cell references for each light ray inside capillary bore were

calculated to allow attribution of light intensity values from each ray to related cells, see Fig.

5. Along each ray path, the light intensity at each cell was calculated as a percentage of initial

intensity from Eq. (2).

The intensity contribution from each ray to each cell was calculated separately and summed

to give the total light intensity value in each cell. Due to the finite size of the cells and finite

distance between incident rays, digitisation of the light intensity values across the capillary

resulted. To visualise a more physically correct result all values were averaged using a 19

point moving average calculation along the x-axis. By using this 19 point moving average

method, there was a loss of accurate information for a 1.9 m region (i.e. 19 m x 0.1 m) at

the outer diameter of the 100 m bore (for x-values from 231 to 250). All numeric

simulations were conducted on custom-built PC with the follow-ing specifications: ASUS

P5E64 Workstation, INTEL Core2Quad Q9450, 4GB OCZ DDR3 PC-10666, and

WindowsTM

7 Enterprise 64 bit operating system. The ray-tracing programme was devel-oped

in National Instruments LabVIEWTM

9.0/2009 Service Pack 1 environment.

4. Results

4.1. Experimental and model results for first macro-simulation tests (set A)

Fig. 7 shows a picture of the laser light (532 nm) passing through the macro-experimental

configuration as per experimental set A. The reflection and transmission angles ( , and

) presented in this picture were those selected for comparison with the numerical model

results.

The results of the angle measurements of , and from these experiments and the

numerical models are presented in Figs. 8-10, respectively. The points in these graphs

represent the average angle values for each given displacement from the centre and the

whiskers on these points show the 95% confidence interval as calculated using the t-

distribution. The solid line shows the corresponding results as calculated by the developed

numerical based software model.

The results of the theoretical model, shown in Figs. 8-10, largely fall within the 95%

confidence intervals for the experimentally measured results and follow similar trends with

respect to increasing values in OR1 and reducing values of an and 0,2 for increasing

displacements. These results indicated good agreement between the model and experimental

measurements for testing with set A conditions, with Pearson correlation coefficients of

0.966, 0.977 and 0.986 for the angle measurements of θ R1, θ T1 and θ T2, respectively.

Significance p-value (2-tailed) in all cases was below 0.001, meaning that one can reject null

hypothesis and should regard data as statistically correlated and a relationship exists.

4.2. Experimental and model results for second macro-simulation tests (set B)

Fig. 11 shows a picture of the laser light (532 nm) passing through the macro-experimental

configuration as per experimental set B. The reflection and transmission angles (yR1, yT1

and yT2) presented in this picture were those selected for comparison with the numerical

model results. The results of the angle measurements of yR1, yT1 and yT2 from these

experiments and the numerical models are presented in Figs. 12–14, respectively. The points

in these graphs represent the average angle values for each given displacement from the

centre and the whiskers on these points show the 95% confidence interval as calculated using

the t-distribution. The solid line shows the corresponding results as calculated by the

developed numerical based software model. The results of the theoretical model, shown in

Figs. 12–14, largely fall outside of the 95% confidence intervals for the experimentally

measured results but do, however, follow similar trends with respect to increasing values in

OR, and On and reducing values of θ_rz for increasing displacements.

The measured angles in Figs. 12 and 13 were, however, found to reasonably well with the

theoretical predictions. Significant differences observed between modelled and measured

values for , see Fig. 14, are most likely due to alignment issues in the experimental set-up.

These factors are considered further in the discussion section. Despite observable

discrepancies between theoretical predictions and experimental results, calculated Pearson

correlation coefficients were 0.997, 0.998 and 0.998 for the angle measurements of

, and , respectively, with significance p-value (2-tailed) below 0.001.

4.3. Experimental and model results for intensity distribution tests

Fig. 15 shows the 3D theoretically calculated light intensity distribution inside half of the 100

m capillary filled with light absorbing solution and its 2D plan view projection. As the

capillary was optically symmetrical and light ray paths could not pass from one half to the

other, only half of the capillary is shown here. The light intensity values recorded in the

model were divided into six ranges as represented by the colour map shown in Fig. 15. The

values from the model were scaled to fall within the range of intensity values from 135 to

255. The values shown against each colour represent the mid-point value for a range of20 on

the colour intensity with values above 255 being assigned within 255-236 range.

This scaling allowed for direct comparison with the experimental results shown in Fig. 16b.

The range of 20 on the colour map represents a span of 7.8% in light intensity values on the

scale of 0 to 255. Light was incident along the y-axis (i.e. from the left side in Fig. 15b). A

series of colour photos were taken of the tartrazine filled 100 Jim inner bore diameter

capillary under illumination of the LEDs. These images were saved as 256 level, grey scale,

"PSD" format images. Such images directly indicate light intensity values. All pixels of

intensity within a ± 10 grey scale range were selected together and a colour was assigned. For

example, level 225 included all pixels within grey scale intensity level range from 216 to 235.

Theoretical predic-tions were compared with these experimentally measured light intensity

maps. Fig. 15a shows the photograph of the light intensity distribution inside a capillary filled

with tartrazine and illuminated with the white LED and Fig. 16b shows the same image

processed into a colour map representing light intensity distribution inside the capillary. Fig.

16a shows the same capillary illuminated with the 430 nm violet LED and Fig. 16b shows the

same image processed into a colour map representing light intensity distribution inside the

capillary. A small shadow effect is present in the upper part of the photographs (Figs. 16a and

17a) as a result of the capillary not being perfectly cut. When com-pared to theoretical

predictions, shown in Fig. 15, similar profiles can be observed.

An interesting feature clearly displayed in the theoretical model is the location of a highest

intensity point at the capillary wall. Its location is a result of capillary focusing, where the

capillary walls act like a lens. Although this point is not directly measured from experiment

(due to a very low detector sensitivity) the layers of crescent-like distributions of the same

intensity concurs well with the theoretical predictions.

5. Discussion

A numerical model of light propagation and intensity distribution for coated fused silica

capillaries has been developed. Theoretical predictions of light paths are concurrent with

previous publications, and are presented with improved capability allowing taking into

account the presence of coating materials and their optical proper-ties. The model itself has

high flexibility allowing calculations for multiple coatings and capillary body materials with

differing dimensions (coating, capillary wall thickness and bore diameters) and optical

properties (refractive indices and light absorptivities). Model has been tested experimentally

and showed a good agreement between theoretical predictions and measured results.

For set A macro-tests, comparison of the experimental and model results for the reflection

and transmission angles ( , and ) showed good agreement between these with model

values largely lying within the 95% confidence intervals of the experimental results. The

results from the models and experi-ments also trended well together. For set B macro-tests,

the , and experimentally measured results were found to match only reasonably well

with the theoretical results. These results were found to largely lie within the confidence

intervals if a 2 mm additional displacement was applied to the incident light. The correlation

coefficients for the set B results were higher than the values for set A indicating the closer

trending of the experimental data with the model values despite higher absolute error.

Reasons for observed discrepancies between theoretical and measured values are most likely

due to experimental error. Much effort was invested to ensure alignment of the individual

components, including mounting the experiment on an optical bench, attaching the laser on a

micrometre stage, which in turn was set-up perpendicular to the edge of the cylinder's base

plate, checking alignment of the reflected beams before measurement for height and

perpendicularity adjustments and mounting the camera on an optical bench stand

perpendicular to the top of the cylinders. However, over the scale of the experiment,

alignment errors could be present due to imperfect alignment of the camera resulting in slight

tilt of the incident laser beam; imperfect alignment of the laser pointer in the x-y plane

resulting in a tilt of the beam; imperfect alignment of the laser pointer parallel to the x-axis of

the micrometric stage; and difficulties in determination of laser beam edges in photographs

resulting in inaccurate angle measurements. A slight shift and tilt of the beam from the true

theoretical position could therefore be present along with some inaccurate edge detection in

used image analysis routines.

From the light intensity distribution simulation an interesting focusing lens behaviour effects

of the capillary were detected. The model clearly displayed the highest intensity location

point at the capillary wall. These simulation results also predicted light intensities of similar

magnitude in crescent-like regions emanating from the capillary walls. These distributions

corresponded well with the light intensity distribution regions as measured experimentally.

The shape of regions of equal intensity, their numerical values and the relative percentage

changes are almost identical between the simulation and experimental results. These

experimental validations show encouragingly good agreement between theoretical predictions

and measured results, which could allow for optimisation of associated regions for monolith

synthesis and use in fluidic chromatography, optical detection systems and flow cells for

capillary electrophoresis and flow injection analysis.

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